Under what single condition does using a blockchain actually make sense, rather than a plain database?

Only when several parties who don't trust each other need to write to a shared system, and none of them will accept an online trusted third party to arbitrate.

A blockchain is, at bottom, just a database with a very expensive extra property: it lets mutually mistrusting writers agree on a shared state without anyone being in charge. That property has real costs (throughput, latency, complexity), so it's only worth paying for when you genuinely can't fall back on something simpler:

• One writer → a normal database is better; there's nobody to distrust.
• Multiple writers who trust each other → a database with shared write access is enough.
• Multiple writers, no mutual trust, and an always-online trusted third party (TTP) they'll all accept → let the TTP run the database; still no blockchain needed.

The blockchain only earns its keep in the leftover case: multiple mutually distrusting writers, and no online TTP everyone agrees on. Everything else is a more expensive way to do what a database already does.

Go deeper:

• Wüst & Gervais, "Do you need a Blockchain?" (IACR ePrint 2017/375) — the primary paper this whole methodology comes from.
• Blockchain (Wikipedia) — background on the "shared append-only ledger" the decision hinges on.